Q-derivative

In mathematics, in the area of combinatorics, the q-derivative is a q-analog of the ordinary derivative.

Definition

The q-derivative of a function "f"("x") is defined as

:$left(frac\{d\}\{dx\}\; ight)\_q\; f(x)=frac\{f(qx)-f(x)\}\{qx-x\}$

It is also often written as $D\_qf(x)$. The q-derivative is also known as the Jackson derivative.

Relationship to ordinary derivatives

Q-differentiation resembles ordinary differentiation, with curious differences. For example, the q-derivative of the monomial is:

:$left(frac\{d\}\{dz\}\; ight)\_q\; z^n\; =\; frac\{1-q^n\}\{1-q\}\; z^\{n-1\}\; =\; [n]\; \_q\; z^\{n-1\}$

where $[n]\; \_q$ is the q-bracket of "n". Note that $lim\_\{q\; o\; 1\}\; [n]\; \_q\; =\; n$ so the ordinary derivative is regained in this limit.

The "n" 'th derivative of a function may be given as

:$(D^n\_q\; f)(0)=frac\{f^\{(n)\}(0)\}\{n!\}\; frac\{(q;q)\_n\}\{(1-q)^n\}=\; frac\{f^\{(n)\}(0)\}\{n!\}\; [n]\; \_q!$

provided that the ordinary "n" 'th derivative of "f" exists at "x"=0. Here, $(q;q)\_n$ is the q-Pochhammer symbol, and $[n]\; \_q!$ is the q-factorial.

See also

* Derivative (generalizations)
* Jackson integral
* Q-exponential
* Q-difference polynomials
* Quantum calculus

References

* Victor Kac, Pokman Cheung, "Quantum Calculus", Universitext, Springer-Verlag, 2002. ISBN 0-387-95341-8
* J. Koekoek, R. Koekoek, " [http://arxiv.org/abs/math/9908140 A note on the q-derivative operator] ", (1999) ArXiv math/9908140